__maths__>__Properties of Vector Arithmetics__>__Properties of Vector Multiplication by Scalar__### Distributive over Scalar Addition

In this page, The property, Multiplication of vector is distributive over scalar addition, is explained.

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What does 'distribute' mean?

- to trouble or disturb
- to give to many
- to give to many

The answer is 'to give to many'.

Considering that a vector has components as real numbers, Given that `vec p = ai+bj+ck` and `lambda + mu` where `a,b,c,lambda,mu in bbb R`, What will be the result of `(lambda + mu) vec p`?

- scalar
- `lambda vec p + mu vec p`
- `lambda vec p + mu vec p`
- `lambda + mu vec p`

The answer is '`lambda vec p + mu vec p`'. This can be proven by multiplying the components individually by the scalar and using vector addition properties.

• A vector multiplied by sum of two scalars equals sum of the vector multiplied by the scalars.

**Distributive Over scalar addition: ** Given scalars `lambda,mu` and vector `vec p`,

`(lambda+mu)vec p = lambda vec p + mu vec p`

*Solved Exercise Problem: *

Given that `vec p = 3 vec q` and `vec r = 2 vec q`, which of the following equals `7 vec q`?

- `vec p + vec r`
- `vec p + 2 vec r`
- `vec p + 2 vec r`
- `vec p - vec r`
- `vec r - vec p`

The answer is '`vec p + 2 vec r`'

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